I do not understand why the second part is in the subjunctive tense. My understanding is that the subjunctive is used for wishes and unreal events but if I understand correctly the write is stating that the eyes are red and watery at the moment and there is nothing unreal about it.

3 Answers
3

From the statement of the question, it is not clear the level of mastering of the French language of the OP, so I have split my answer starting from some preliminary notes going up to a specialized publication.

General Remarks

There are some subordinating conjunctions that require the
subjunctive mood. Among them is bien que. Subordinating conjunctions,
are used to join a subordinating clause (dependent clause) to main
clauses. In particular, bien que belongs to the so called opposition
conjunctions that include also malgré que, sauf que, sans que...

When bien que is not a conjunction

When bien que does not imply opposition it does not require subjunctive.

Je pense bien qu'il t'aime.

Here bien que is an adverbial preposition.

An ongoing debate

Regarding bien que as conjunction there is an ongoing debate between
those stating that one must use the subjunctive and others who state
that bien que may be followed by the indicative according to the context. See for instance
here

(The reality of the
event may well be accepted, but it is discounted – e.g. in the second
of these examples it is accepted that the letter has been received,
but in spite of that fact, the brother is not coming.)

Bien que and
quoique occasionally take the indicative or conditional when ‘though’
is almost the same as ‘but’ e.g.:

ses yeux sont verts, bien que, pour l'instant, il sont rouges exactly as you said it in english. You are correct.

It should however be stated that (and, it seems particularily needed to insist here) in the particular case of rectifying concessionsmode is absolutely not constrained.
While being himself responsible for both judgements, the locutor might want to insist on the correction, on the impact of P2 on P1 and keep indicative for P2. Or... contrarily, show a simple and temporary discontinuity in the narration plan.

So, in order to precisely answer your question : Why is subjunctive being used ?

I suspect that is the reason why the author chose the subjonctive here, P2 does not fundamentally / definitely contradicts P1. P2's value is just... accidental.

C'est pas pour moi:-)! The question states : "I do not understand why the second part is in the subjunctive tense. My understanding is that the subjunctive is used for wishes and unreal events..." I thought that the user does not know about the conjunctions that require the subjonctive, therefore, my response should cover the fundamentals. See all my references. All state clear that bien que (concession) requires subjonctive. I understand your point of view, I thank you for your kind comments regarding my level of French, I can follow the debate here...
– DimitrisDec 31 '18 at 13:21